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Physica Scripta

Generation of exactly solvable quantum potentials in D-dimensional space

N Saikia, Syed Afroz Ahmed

Published Mar 1, 2010• 14 citations
Publisher

Abstract

Exactly solved quantum potentials are obtained using the extended transformation technique within the framework of Green's functions, in any chosen D-dimensional space. Bound state solutions of the exactly solved potentials are given. It is shown that the bound state eigenfunctions of the generated quantum system can be easily normalized in most cases.

DOI
10.1088/0031-8949/81/03/035006

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Exactly solved quantum potentials are obtained using the extended transformation technique within the framework of Green's functions, in any chosen D-dimensional space. Bound state solutions of the exactly solved potentials are given. It is shown that the bound state eigenfunctions of the generated quantum system can be easily normalized in most cases.
A transformation method is presented which consist of a coordinate transformation and a functional transformation that allow generation of exact analytic bound state solutions of the Schrodinger Green's function equation. The method proposed here, which involves the generation of an exact analytic solution from solvable decemvirate power potential within the framework of Green's functions technique, makes it possible to generate a number of solved quantum systems for original quantum systems with multi-term potentials. The generated quantum systems are in general energy dependent with a single normalized eigenstate. A method has been devised to convert a subset of the generated quantum systems with energy-dependent potentials to a single normal system with an energy-independent potential.
Exactly solvable models play an extremely important role in many fields of quantum physics. In this study, the Schrödinger equation is applied for a solution of a two--dimensional (2D) problem for two particles interacting via Kratzer, and modified Kratzer potentials. We found the exact bound state solutions of the two--dimensional Schrödinger equation with Kratzer--type potentials and present analytical expressions for the eigenvalues and eigenfunctions. The eigenfunctions are given in terms of the associated Laguerre polynomials.
A simple technique employed almost three decades ago to manufacture partially solvable quantum many-body problems is revisited. [A quantum problem is “partially solvable” if (only) some of its eigenvalues and eigenfunctions can be exhibited]. The models thereby generated are characterized by Hamiltonians of normal form, i.e., standard kinetic plus momentum-independent potential energy; in most cases the latter features three-body, in addition to two-body and one-body, interactions. The setting refers to D-dimensional space; the examples focus on D=1, D=2, and D⩾2, and include generalizations of, and additional results on, cases recently discussed in the literature, as well as new models.
We extend the notion of Dirac oscillator in two dimensions, to construct a set of potentials. These potentials become exactly and quasi-exactly solvable potentials of nonrelativistic quantum mechanics when they are transformed into a Schrödinger-like equation. For the exactly solvable potentials, eigenvalues are calculated and eigenfunctions are given by confluent hypergeometric functions. It is shown that, our formulation also leads to the study of those potentials in the framework of the supersymmetric quantum mechanics.
In this article, using an exactly‐solvable multiparameter exponential‐type potential we propose a unified treatment of the analytical bound—state solutions of the Schrödinger equation for exponential‐type potentials in D‐dimensions. Our proposal accepts different approximations to the centrifugal term; however, its usefulness is exemplified in the frame of the Green and Aldrich approach. This fact enables us to compare our results with specific potentials found in the literature and that are obtained here as particular cases of our proposal. That is, instead of solving a specific exponential‐type potential, by resorting each time to a specialized method, the energy spectra and wavefunctions are derived straightforward from the proposed approach. Furthermore, our proposal can be used as an alternative way in the search of solutions to new exponential‐type potentials besides that one can study different approximations to the term . © 2014 Wiley Periodicals, Inc.